How to Change the Base of a Logarithm

The change of base rule converts a logarithm in a given base to a logarithm in a new base. The result will be a quotient (fraction) with the new logarithm as both the numerator and denominator.

The change of base rule for logarithms

For example, convert to logarithm base 5.

Using the change of base formula, , a=3, b=81 and the new base c=5.

The base is changed to base 5 like so: .

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As long as the new base of the logarithm chosen is the same on the numerator and denominator of the fraction, any base can be chosen.

For example, can also be written in base 10 in the same manner using .

Both and are equal to .

When to Use the Change of Base Formula

The change of base formula is used to rewrite a logarithm so that the logarithm has a new base. This is useful for evaluating logarithms on a calculator when only log base 10 is available. It can also be useful for simplifying some logarithmic expressions.

Some calculators do not have the function for entering different bases and simply have a ‘log’ button. This means that these calculators can only work in base 10.

For example, we previously considered . If a calculator does not have a function for entering log base 3, then this cannot be calculated.

If the calculator has the ‘log’ button, then the base can be changed to base 10 and then evaluated. The ‘log’ button means log base 10.

Evaluating Logarithms Using the Change of Base Formula

To evaluate a logarithm using the change of base formula, use log_a(b)=log(b)÷log(a). The calculation of log(b)÷log(a) can be done on a calculator to work out log_a(b).

The change of base formula is .

By choosing the new base to be 10 and writing the fraction as a division, the change of base formula can be more easily evaluated with the following rule:

Change of base formula for evaluating logs

For example, evaluate .

Here a=3 and b=10.

Using with these values, .

Evaluating this on a calculator,

Here is another example of evaluating a logarithm using the change of base formula.

Using with these values, .

On a calculator, .

Change of Base Formula with Natural Log

The change of base formula also works with the natural logarithm, ln. To change the base of a logarithm to a natural logarithm, use the formula: log_a(b)=ln(b)/ln(a).

Natural logarithms can be evaluated on a calculator. Therefore .

Proof of the Change of Base Formula for Logs

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1. Let log_ab = y such that a y =b.
2. Take log_c of both sides such that log_ca y =log_cb.
3. Bring down the power of y in front of the log to get ylog_ca=log_cb.
4. Divide both sides by log_ca to get y=[log_cb]/[log_ca].
5. Substitute y=log_ab such that log_ab=[log_cb]/[log_ca].

How to Change the Base of a Log on a Calculator

To find the logarithm of a different base on a calculator, use the change of base button. Most calculators have the option to enter the base of a logarithm. This allows any logarithm to be evaluated after using the change of base formula.

Changing the base to base 5, .

Evaluating this on a calculator .

To enter logs with different bases on a Ti-84 calculator:

1. Press the [math] key
2. Scroll down and select ‘logBASE(‘ from the list

How to Change the Base of Exponents

To change the base of an exponent from base ‘a’ to base ‘c’, use the formula a b =c b⋅log_c(a) .

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The change of base formula for exponents

For example, write 3 5 as a power of 2.

Here, a=3, b=5 and c=2.

We can approximate and so

The change of base rule for exponents can be used to write any exponential as base e, where ‘e’ is Euler’s number, .

For example, write 5 x as an exponential with base e.

Here a=5, b=x and c=e.

Since log base e is the natural logarithm, ln, this can be written as .

Any exponential equation can be written as an exponential equation with base e using the formula: a x =e xln(a) .